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I am looking for a nice answer to the following question.

Which immersed plane curves bound an immersed disk?

Comments.

  • I am not sure what is a nice answer, but for sure I could make a stupid algorithm.

  • I am aware that there are plane curves which bound few "different" immersed disks. For example this Bennequin’s curve is bounding five different immersed disks. It suggests that there is no "nice" answer to the question.

Bennequin's curve

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The answer was given by Samuel Blank in his Brandeis 1967 phd dissertation, on which Poenaru gave a Bourbaki seminar.

Then Peter Shor and C. J. Van Wyk gave a polynomial time algorithm to decide if there is an extension.

EDIT: Blank's method already gave a polynomial algorithm, but with an exponent too large to make it practical, which is needed for applications (for instance to integrated circuit design).

The answer is in term of existence of a chain of "reductions" of a certain kind for the cyclically reduced word in the free group $F_n$ on $n$ generators determined by the immersion, where $n$ is the number of bounded components of the complement of the curve (assumed to have only transverse double points).

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    $\begingroup$ I had seen may references to Blank's thesis, but had not been able to find it. Thanks for the reference to Poenaru !! $\endgroup$
    – alvarezpaiva
    Jun 21, 2012 at 15:21
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    $\begingroup$ Some related algorithmic / complexity questions are treated in this paper of Eppstein and Mumford arxiv.org/abs/0806.1724 . $\endgroup$
    – j.c.
    Oct 13, 2017 at 14:01
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Dear Anton,

Some time ago I was looking at this question (which is important for Chekanov's invariants of legendrian links) and the literature is rather scattered (however, look at the work of Charles Titus in MathSciNet).

I found the following paper, but I haven't really taken a good look at it yet.

"When Does a Curve Bound a Distorted Disk?

Jack E. Graver and Gerald T. Cargo Consider a closed curve in the plane that does not intersect itself; by the Jordan–Schoenflies theorem, it bounds a distorted disk. Now consider a closed curve that intersects itself, perhaps several times. Is it the boundary of a distorted disk that overlaps itself? If it is, is that distorted disk essentially unique? In this paper, we develop techniques for answering both of these questions for any given closed curve in the plane.

Read More: http://epubs.siam.org/doi/abs/10.1137/090767716"

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